Theorem ax12o 1934

Description: Derive set.mm's original ax-12o 2142 from the shorter ax-12 1925. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.)

Assertion

Ref	Expression
ax12o	|- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))

Proof of Theorem ax12o

Dummy variables w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax12v 1926	. . 3 |- (-. z = y -> (y = w -> A.z y = w))
2		ax12v 1926	. . 3 |- (-. z = y -> (y = v -> A.z y = v))
3	1, 2	ax12olem4 1930	. 2 |- (-. z = y -> (-. A.z -. y = w -> A.z y = w))
4		ax12v 1926	. . 3 |- (-. z = x -> (x = w -> A.z x = w))
5		ax12v 1926	. . 3 |- (-. z = x -> (x = v -> A.z x = v))
6	4, 5	ax12olem4 1930	. 2 |- (-. z = x -> (-. A.z -. x = w -> A.z x = w))
7	3, 6	ax12olem7 1933	1 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax12  1935  dvelimv  1939  hbae  1953  nfeqf  1958  dvelimh  1964  dvelimf  1997  dvelimALT  2133  ax11eq  2193  ax11indalem  2197  axi12  2333